The Function sin x/x
Author(s): William B. Gearhart and Harris S. Shultz
Reviewed work(s):
Source: The College Mathematics Journal, Vol. 21, No. 2 (Mar., 1990), pp. 90-99
Published by: Mathematical Association of America
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sinx
The
Function
William B. Gearhart
Harris S. Shultz
William B. Gearhart received his B.S. degree in engineering
physics from Cornell University,and his Ph.D degree in applied
mathematics, also fromCornell University.He is currentlyPro?
fessor of Mathematics at the California State University,Fullerton. His research interests include approximation theory, numer?
ical analysis, optimization theory, and mathematical modeling.
Harris S. Shultz, Professor of Mathematics at California State
University,Fullerton, is the author of more than fortypublished
papers and a textbook, Mathematical Topics for Computer
Instruction.He has been Co-Principal Investigatorof the South?
ern California Mathematics Honors Institute,the Mariana Islands
Mathematics Institute,and the C3 Teacher Training Project, all
funded by the National Science Foundation. In 1988 he was
named Outstanding Professor at California State University,
Fullerton, and in 1989 he was the recipient of the California
State UniversityTrustees Outstanding Professor Award.
Calculus students are well aware of an important use of the function sin x/x. The
common and perhaps the easiest approach to finding the derivatives of the trigono?
metric functions hinges on the result
sin*
lim--1.
x->0 X
(i)
However, few students realize that this function plays a key role in many areas of
mathematics and its applications. In this paper, we briefly describe several of these
arises
roles after first presenting some elementary examples in which sin*/*
are
accessible
to
calculus
students
and
These
may
help
examples
geometrically.
motivate interest in and draw attention to this function. In addition, they provide
some unusual
Properties
proofs of the limit (1).
of sin x/x
In certain fields, such as signal analysis, the function sinx/jc
"sine function." Thus, let us set
sine x =
is often called
the
sinx
= 1, the function is extended to an analytic function on the real
By defining sincO
line. We shall refer to this extension also by the name sine. The graph of sine is
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shown in Figure 1, where we note that it is an even function with roots at ?tt, for
= 0.
n = ? 1, ? 2,..., that |sincx| < 1 for all x ?> 0, and that lim^
?00 sine*
1 sincx
The sine function has a rapidly convergent power series representation,
An
sincx=
? (-i)"___
(2#i + l)!
?_o
and even an infinite product representation [7],
?
sine x = _,
J~[ j 1
2 2
Also,
/oo
sincxrfx = 7r,
although this integral is not absolutely convergent. However,
integrable on the real line with
/oo
the square of sine is
sine xdx = m.
All things considered, sine is a rather nice, well-behaved function. Properties of
sine are presented in [4, sections 39-41] and [2, pages 62-63].
Examples
from Geometry
In this section we present a few simple examples from geometry involving sin x/x.
In the process, some nonroutine proofs of the limit (1) will fall out.
In a circle of radius 1,
Area off a circle.
vertex angle a and one
with
isosceles triangles
=
<
<
<
m
a
where 0
(Figure 2). Note that j8
fl
sides
unit
with
area of an isosceles triangle
inscribe a polygon composed of n
isosceles triangle with vertex angle /?
2<n - na and that lima^ 0+j3 = 0. The
It
and vertex angle 0 is (l/2)sin0.
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91
Figure 2
follows that the area of the polygon is
A(a)
=
Hasina
277-0
=-?sin
a
4- \sinP
1
2
1
a 4- ?sin B.
2
277-0
= ?-?sine
2
Hence,
1
a + ? sin/?
2
a proof of (1) follows from
77=
lim A(a)
a?0+
277-0
= ?-?
2
lim sine a 4- 0.
a-+0
Further, the perimeter of the polygon is
P(a)
/
P
a\
= nUsin4-2sin-=
= (277 ? jS)sinc?
v
'
a
2
(2ir-P\(
-
2sin-
a\
4-2sin-
P
P
+ 2 sin?.
2
As a approaches zero, the perimeter approaches 277, giving us another verification
of the limit (1).
It is interesting to note that some early attempts to measure 77 were based on
measuring the perimeter of a regular polygon inscribed in a circle. For a regular
=
= 0. From the last formula, the perimeter is
2ir/n, and 0
polygon with n sides, a
277 sinc(77/?). Thus, the early mathematicians were actually using 77sinc( 77/72)as an
approximation of 77.
The centroid, or center of gravity, of a plane region is the
Centroid of a sector.
point at which a fulcrum should be placed so that the region balances. Consider a
circular sector having radius 1 and central angle 2 a, drawn in Figure 3 so that the
vertical axis is the angle bisector. The centroid of the circular sector lies on this
vertical axis. The moment about the horizontal axis through the vertex is given by
fy/l^
= ? sina.
ydy dx
r -sinaV^-xcota
(/
-5
/
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(- sin a, cos a)
Figure 3
Since the area of the sector is ?, the distance to the centroid from the vertex is
y
=
(2/3) sin a
2
= ?sine a.
3
can now give a physical argument to justify the limit (1). Suppose two
triangles are drawn as illustrated in Figure 4. The centroid of an isosceles triangle is
on its altitude 2/3 of the way from the vertex. Therefore, the centroid of the larger
triangle is 2/3 units above the vertex, and the centroid of the smaller triangle is
(2/3)cos a units above the vertex. Since the centroid of the sector must lie between
those of the triangles, we have
We
?cos
3
a < ?smc
3
a < ?.
3
Thus, allowing a to approach zero gives the result (1).
Another example involving sine and centroids appears in the recent paper of
Lo Bello [10]. Consider an w-sided pyramid whose base is a regular w-sided polygon
inscribed in a circle of radius r and whose peak is at a height h above the center of
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93
the circle. As shown by Lo Bello [10, p. 169], the volume of this pyramid and the
moment about its base are
V=
1
277
,
?TTr2h sine?
n
3
Note that as n approaches
and
1
=
MYW
xy ?mr
12
277
h sine?.
/i
infinity, the formulas become those for the cone.
In Buffon's needle problem, a needle L units long is
Buffon's needle problem.
thrown at random onto a plane ruled with horizontal parallel lines d units apart
(Figure 5). Assume L < d and let a be a fixed angle such that 0 < a < m/2. We seek
the probability that the needle crosses a line, given that it lies within an angle a of
the vertical. (In the usual case of Buffon's needle problem, a = it/2.)
Figure 5
Let C be the event the needle crosses a line, and let A (a) be the event the needle
lies within an angle a of the vertical. If y is the distance of the southernmost point
of the needle from the line immediately above, and 0 is the angle the needle makes
to occur we must have both
with the vertical, then in order for the event CC\A(a)
y < L cos 0
? a < 0 < a.
and
Thus,
rLcosO
P{CnA{a)\
/? J
= -^-^-=
dyd0
a
2L
\
?- f LcosOd0=?since.
"d
*dJ-?
r/2 [ddyd0
J
-tt/2J0
But the probability of the event A(a) is 2a/77. Hence, the conditional probability of
the event C, given that the event A (a) occurs, is
P{C\A(a)}=
2L
??-sin<
md
wdla
a
r
^
=^sinca.
77
If d = L, then the probability that the needle crosses a line, given that it lies
within an angle a of the vertical, is simply sine a. As a approaches
0, this
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probability must approach 1. Thus, we have a probabilistic interpretation of the
limit (1). For this argument to be a proof, however, one must evaluate the integral
above without implicitly using the limit (1). The article by Cipra [S] indicates a way
of finding JcosOdB without knowing the limit.
Expressions
for ir
The sine function has played a role in the search for analytical expressions for the
universal constant 77. The recent article by Castellanos [3] gives a fascinating review
of the quest. One of the firstexpressions for 77,due to Francois Vieta in 1593, can be
obtained by observing that
x
sin?
x
xx
2 =
cos? ?y?
cos?sine?
2
22
2_
2
sinx
sine* =-=
x
xxx
= cos?cos?sine?,
24
4
and so, for arbitrary ?,
?A*
sine* = cos?cos?
2
*\
4
?/V
?A*
? ? ? cos?-sine-?-.
2"
2"
Using the limit (1), we get
sincx=
x
A
ricos^>
which is known as Euler's formula. Now, we take x = 77/2, and use the half-angle
formula for cosine to obtain the expression
rnn+m
This is the result due to Vieta [3, p. 69]. Evaluating this expression would be a good
calculator exercise, but unfortunately the convergence is very slow.
The following formula for 77 also was obtained by Euler:
772
?
1
2'
To get this result, Euler equated
the coefficient of x2 in the power series for sine,
sinc* = l--
X2
X4
X6
+ ---+-..
to the coefficient oi x2 in the product expansion
Sine* -
1 - -r-r
1 - -r-r
132772
127T2]\
22772A
Euler obtained yet another formula for 77 in this way by equating coefficients of the
fourth powers of x in these expansions [3, p. 75].
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95
Some
Roles
offsinx/x
We mention now a few of the classical ways in which sine occurs in mathematics
and its applications. This survey, together with the references, will provide the
reader with direction for further study.
The sine function appears frequently in Fourier analysis, a
analysis.
major technique in the solution of differential equations and in other applications of
mathematics. For a periodic function / that has period 277 and satisfies other
reasonable conditions [4, page 90], Fourier analysis allows us to expand the function
in a trigonometric series of the form
Fourier
f(x)
=
a0 + axcosx
4- b1sinx
4- a2 cos 2.x 4- 62 sin 2.x 4- ? ? ?,
(2)
where the a/s and the Z>/s are constants that can be determined.
A function on the real line that is nonperiodic cannot be expressed as a
trigonometric series. However, the Fourier integral formula gives an expression for
such a function involving integrals of sines and cosines (rather than sums) that
converges to the function at points of continuity. To derive the Fourier integral
formula, we first need the following result, which can be found in [6, p. 78]: If / is a
function on the real line that is continuous at the point x and satisfies certain other
reasonable conditions, then for any positive number a,
lim f
af(t)-sincs(t-x)dt.
-a
77
s -? oo Jx
f(x)=
(3)
This result leads immediately to the Fourier integral theorem as follows. First note
that
1 rs
sin ux
cos cox dw ?77 ^0
7TX
s
= ? sines*.
77
We can then write
1 t-s
px+ a
I cos[co(tlim /
f(t)?
77?'o
s -+oo Jx- a
f(x)=
x)]dwdt.
Changing the order of integration and extending the integration with respect to t
from - oo to oo [6, p. 78], we get
/(*)
=
1 p<x>/?oo
-/
/ /(Ocos[?(f-x)]
77^0 ?'-oo
dtda,
which is the Fourier integral formula. This derivation follows [6]; see also [4] and
[1], particularly for proofs of (3). The Fourier integral formula can also be written in
exponential form [1]:
/(*)-7-f? ^77 ^-oo^-oo
f? f(t)e-'ui,-x}dtdU,
where i is the imaginary number ]/? 1 and elt = cos 14- i sin t. We can then write
/(*)
96
=
?rF(?y?*rf6>,
277 ^-oo
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where
f(t)e~Mdt.
/oo
-oo
This last integral is called the Fourier transform of /.
From formula (3) we see that the sine function lies at the core of the Fourier
integral formula. However, sine appears in other ways in Fourier analysis. A
noteworthy example is its role in smoothing Fourier series to improve convergence
and treat Gibbs' phenomenon. Gibbs' phenomenon is the tendency of a truncated
Fourier series to display oscillations near points of discontinuity. Smoothing or
averaging is done to reduce this effect. To smooth a Fourier series truncated to n
terms, we simply multiply the fcth term by the factor sinc(A:77/?) [9, pages
530-538].
As noted by Courant and Hilbert [6, p. 78], the property of sine expressed in
formula (3) was discovered by Dirichlet, and this formula is often used as the
foundation of the theory of Fourier series. In certain fields where Fourier methods
are fundamental, such as image processing, sine is appropriately known as the
Dirichlet function.
The appearance of sine in Fourier analysis explains its occur?
Spectral
theory.
rence in areas of engineering that rely on the analysis of signals, such as communi?
cation theory and acoustics. Mathematically, a signal is simply a function of time.
In practice, a signal may actually be a voltage or current, which in turn represents
another physical quantity such as sound energy. For a periodic signal / with
Fourier series (2), we can view the signal as "decomposed"
into periodic functions
and "amplitudes" given by the absolute values
with discrete frequencies 0,1,2,...,
of the coefficients a0, aly bv....
For a nonperiodic signal / on the real line we must
allow every frequency co. In the Fourier integral representation
1
yOO
2tt J-oo
we can view / as decomposed into a continuous "spectrum" of periodic functions of
x, F(o))eiuiX, for the various frequencies to. The analogue of the amplitudes in (2) is
\F(ci)\,the absolute value of the Fourier transform. One can interpret \F(o>)\ as an
"index of the relative amplitudes of the components of the frequency spectrum of
/" [11, p. 181]. In fact, \F(o))\2, called the power spectrum of / at <o [2], [9],
measures the intensity of the component with frequency <o.
A fundamental signal is the rectangular pulse which has the form
(t)=(S/2h
//,U
\0
if -h<t<h
otherwise
for positive constants S and h. The integral over time of a force acting on a system
is a measure of the strength of the force. Hence, this rectangular pulse represents a
force of strength S applied for a time 2/z. The Fourier transform of fh(t) is
**(?)=
hy '
r X7e~i0)tdt=-ZT7-r(e-i?h-ei?h)=Ssinc(oh.
J-h2h
2h(-io))K
'
From the graph of sine in Figure 1, we see that for fixed h, the larger frequencies co
are weakly represented in the signal fh. However, since limJC_>0sincx = 1, it also
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97
follows that for any given frequency co, as h -> 0 the component of the signal with
frequency <o becomes stronger. In fact, the smaller the duration h, the larger the
range of the frequencies with nearly the same strength S. In acoustics, a pulse of
high strength and small duration can be realized, for example, when two hard
spheres collide at a high speed. In this case, all the frequencies in the audible range
would be represented with nearly uniform strength and a sharp click is heard.
Perhaps the most celebrated result in communication theory is Shannon's sam?
pling theorem. A function is said to be band limited with band width fll if its
Fourier transform vanishes (F(co)-O)
outside the interval (-S, Q). Band limited
functions are common as many devices effectively transmit only a certain range of
frequencies. Shannon's theorem states that for a band limited function /,
/(*)-
?
?
/(
jsinc
77(20*-*).
Proofs can be found in Hamming [9, page 557] and Rosie [13, page 53]. This
remarkable result states that by sampling a band limited function only at the
1
2
we can reconstruct the entire function.
discrete set of points < 0, ? ?,
? ?,...),
2\l
2\l
j
\
Further, as Rosie states [13, p. 56], the function is "reconstituted by erecting at each
sampling point a sin x/x function of magnitude equal to the sampled value at that
point."
Further reading about the roles of sine in the field of signal analysis appear in [2],
[11], [12], and [13].
Approximation
Long before the sampling theo?
theory and numerical analysis.
rem of Shannon appeared, E, T. Whittaker [15] noticed many important properties
of the sine function, and saw its role in the approximation of functions. Suppose the
values of a function / on the real line are given at a discrete set of points nh,
n = 0, + 1, ? 2,...,
for h > 0. Then / can be interpolated at these points by the
function
C(x)=
J2
/(/i/i)sinc771
? - n\
wherever the series converges. Since the roots of sine are the nonzero multiples of 77,
and sincO = 1, the function C takes on the same values as / at the points nh9 n = 0,
n = ? 1,... . The function C is called the Whittaker cardinal function for /, or the
"sine function" expansion of /. From Shannon's sampling theorem, we see that
Whittaker's cardinal function will reproduce a band limited function when the
= 1/(20),
The function C has been used in
spacing between data points is h
applications to the transmission of information [14, p. 166]. Moreover, Stenger [14],
in an excellent article, analyzes many significant applications of the Whittaker
cardinal function throughout numerical analysis, including the areas of interpola?
tion theory, numerical integration, and solution of differential and integral equa?
tions. As Stenger notes, E. T. Whittaker used C to obtain alternate expressions of
entire functions, and called it ua function of royal blood in the family of entire
functions, whose distinguished properties separate it from its bourgeois brethren."
We have spent some time studying sin x/x and its several roles in
Conclusion.
mathematics and applications. The results of the previous section hint that, in some
sense, it is a basic building block for a large class of functions. There are other
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occurrences of sine that we could have discussed. In mechanics and quantum
physics, it arises in the solution of the wave equation [8, section 16.3]. In probability
theory, sine is the characteristic function of the uniform random variable on (? 1,1).
We hope this article will stimulate readers to think about the prevalence of the
sine function. When we introduce sin x/x in the calculus course, preparing to find
the derivatives of sin x and cos x, we might point out that this function is significant
in its own right, and that its use in calculus is not just an isolated occurrence. Many
of our calculus students are engineering and science majors, and are likely to see this
function again.
References
1. T. Apostol, Mathematical
Analysis,second edition,Addison-Wesley,
Reading,MA, 1974.
and Its Applications,
secondedition,revised,McGraw-Hill,New
2. N. Bracewell,The FourierTransform
York, 1986.
3. D. Castellanos,The ubiquitousit, MathematicsMagazine 61 (1988) 67-98 and 61 (1988) 148-163.
4. R. V. Churchill,FourierSeries and BoundaryValue Problems,second edition,McGraw-Hill,New
York, 1963.
Journal18 (1987)
5. B. A. Cipra, The derivativesof thesine and cosine functions,
CollegeMathematics
139-141.
6. R. Courant and D. Hilbert,Methodsof MathematicalPhysics,vol. I, IntersciencePublishers,John
Wiley& Sons, New York, 1962.
7 . R. Courant and F. John,Introduction
to Calculusand Analysis,volumeI, IntersciencePublishers,
JohnWiley & Sons, New York, 1965, p. 602.
to QuantumMechanics,Addison-Wesley,
8 . R. H. Dicke and J. P. Wittke,Introduction
Reading,MA,
1960, pp. 297-300.
9 . R. W. Hamming, NumericalMethodsfor Scientistsand Engineers,second edition,McGraw-Hill,
New York, 1973.
10. A. J. Lo Bello, Volumes and centroidsof some famousdomes, MathematicsMagazine 61 (1988)
164-170.
11. N. W. McLachlan, Complex VariableTheoryand Transform
Calculus,second edition,Cambridge
UniversityPress,Cambridge,1963.
12. C. W. McMullen, Communication
TheoryPrinciples,Macmillan,New York, 1968.
and Communication
13. A. M. Rosie, Information
Theory,second edition,Van Nostrand Reinhold,
London, 1973.
14. F. Stenger,Numericalmethodsbased on theWhittaker
cardinal,or sine functions,SIAM Review23
(1981) 165-224.
15. E. T. Whittaker,On the functionswhich are representedby the expansionsof the interpolation
35 (1915) 181-194.
theory,Proc. Roy. Soc. Edinburgh
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